A factory owner must decide which of two alternative electric generator systems should be installed in their factory. If each generator is tested 175 times and the first generator fails to work (does not start or does not transmit electricity) 15 times and the second generator fails to work 25 times, test at the 0.01 level of significance whether the difference between the corresponding sample proportions is significant, using

(a) the \(\chi^{2}\) statistic on page 312 ;

(b) the \(Z\) statistic on page 314 .

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EXAMPLE 8 Testing the equality of three proportions using the x statistic Samples of three kinds of materials, subjected to extreme temperature changes, pro- duced the results shown in the following table: Material A Material B Material C Total Crumbled 41 27 22 90 Remained intact 79 53 78 210 Total 120 80 100 300 Use the 0.05 level of significance to test whether, under the stated conditions, the probability of crumbling is the same for the three kinds of materials.